Calculus Examples

$y = - \frac{1}{4} x^{- 4} - \frac{1}{16} + \frac{1}{4} x^{4}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $- \frac{1}{4} x^{- 4} - \frac{1}{16} + \frac{1}{4} x^{4}$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{4} x^{- 4}] + \frac{d}{d x} [- \frac{1}{16}] + \frac{d}{d x} [\frac{1}{4} x^{4}]$ .

$\frac{d}{d x} [- \frac{1}{4} x^{- 4}] + \frac{d}{d x} [- \frac{1}{16}] + \frac{d}{d x} [\frac{1}{4} x^{4}]$

Step 1.2

Evaluate $\frac{d}{d x} [- \frac{1}{4} x^{- 4}]$ .

Tap for more steps...

Step 1.2.1

Since $- \frac{1}{4}$ is constant with respect to $x$ , the derivative of $- \frac{1}{4} x^{- 4}$ with respect to $x$ is $- \frac{1}{4} \frac{d}{d x} [x^{- 4}]$ .

$- \frac{1}{4} \frac{d}{d x} [x^{- 4}] + \frac{d}{d x} [- \frac{1}{16}] + \frac{d}{d x} [\frac{1}{4} x^{4}]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 4$ .

$- \frac{1}{4} (- 4 x^{- 5}) + \frac{d}{d x} [- \frac{1}{16}] + \frac{d}{d x} [\frac{1}{4} x^{4}]$

Step 1.2.3

Multiply $- 4$ by $- 1$ .

$4 (\frac{1}{4}) x^{- 5} + \frac{d}{d x} [- \frac{1}{16}] + \frac{d}{d x} [\frac{1}{4} x^{4}]$

Step 1.2.4

Combine $4$ and $\frac{1}{4}$ .

$\frac{4}{4} x^{- 5} + \frac{d}{d x} [- \frac{1}{16}] + \frac{d}{d x} [\frac{1}{4} x^{4}]$

Step 1.2.5

Combine $\frac{4}{4}$ and $x^{- 5}$ .

$\frac{4 x^{- 5}}{4} + \frac{d}{d x} [- \frac{1}{16}] + \frac{d}{d x} [\frac{1}{4} x^{4}]$

Step 1.2.6

Move $x^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{4}{4 x^{5}} + \frac{d}{d x} [- \frac{1}{16}] + \frac{d}{d x} [\frac{1}{4} x^{4}]$

Step 1.2.7

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.2.7.1

Cancel the common factor.

$\frac{4}{4 x^{5}} + \frac{d}{d x} [- \frac{1}{16}] + \frac{d}{d x} [\frac{1}{4} x^{4}]$

Step 1.2.7.2

Rewrite the expression.

$\frac{1}{x^{5}} + \frac{d}{d x} [- \frac{1}{16}] + \frac{d}{d x} [\frac{1}{4} x^{4}]$

Step 1.3

Since $- \frac{1}{16}$ is constant with respect to $x$ , the derivative of $- \frac{1}{16}$ with respect to $x$ is $0$ .

$\frac{1}{x^{5}} + 0 + \frac{d}{d x} [\frac{1}{4} x^{4}]$

Step 1.4

Evaluate $\frac{d}{d x} [\frac{1}{4} x^{4}]$ .

Tap for more steps...

Step 1.4.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4} x^{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{x^{5}} + 0 + \frac{1}{4} \frac{d}{d x} [x^{4}]$

Step 1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1}{x^{5}} + 0 + \frac{1}{4} (4 x^{3})$

Step 1.4.3

Combine $4$ and $\frac{1}{4}$ .

$\frac{1}{x^{5}} + 0 + \frac{4}{4} x^{3}$

Step 1.4.4

Combine $\frac{4}{4}$ and $x^{3}$ .

$\frac{1}{x^{5}} + 0 + \frac{4 x^{3}}{4}$

Step 1.4.5

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.4.5.1

Cancel the common factor.

$\frac{1}{x^{5}} + 0 + \frac{4 x^{3}}{4}$

Step 1.4.5.2

Divide $x^{3}$ by $1$ .

$\frac{1}{x^{5}} + 0 + x^{3}$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Add $\frac{1}{x^{5}}$ and $0$ .

$\frac{1}{x^{5}} + x^{3}$

Step 1.5.2

Reorder terms.

$f' (x) = x^{3} + \frac{1}{x^{5}}$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Differentiate.

Tap for more steps...

Step 2.1.1

By the Sum Rule, the derivative of $x^{3} + \frac{1}{x^{5}}$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [\frac{1}{x^{5}}]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [\frac{1}{x^{5}}]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [\frac{1}{x^{5}}]$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{1}{x^{5}}]$ .

Tap for more steps...

Step 2.2.1

Rewrite $\frac{1}{x^{5}}$ as ${(x^{5})}^{- 1}$ .

$3 x^{2} + \frac{d}{d x} [{(x^{5})}^{- 1}]$

Step 2.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{5}$ .

Tap for more steps...

Step 2.2.2.1

To apply the Chain Rule, set $u$ as $x^{5}$ .

$3 x^{2} + \frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{5}]$

Step 2.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 1$ .

$3 x^{2} - u^{- 2} \frac{d}{d x} [x^{5}]$

Step 2.2.2.3

Replace all occurrences of $u$ with $x^{5}$ .

$3 x^{2} - {(x^{5})}^{- 2} \frac{d}{d x} [x^{5}]$

Step 2.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$3 x^{2} - {(x^{5})}^{- 2} (5 x^{4})$

Step 2.2.4

Multiply the exponents in ${(x^{5})}^{- 2}$ .

Tap for more steps...

Step 2.2.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 x^{2} - x^{5 \cdot - 2} (5 x^{4})$

Step 2.2.4.2

Multiply $5$ by $- 2$ .

$3 x^{2} - x^{- 10} (5 x^{4})$

Step 2.2.5

Multiply $5$ by $- 1$ .

$3 x^{2} - 5 x^{- 10} x^{4}$

Step 2.2.6

Multiply $x^{- 10}$ by $x^{4}$ by adding the exponents.

Tap for more steps...

Step 2.2.6.1

Move $x^{4}$ .

$3 x^{2} - 5 (x^{4} x^{- 10})$

Step 2.2.6.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x^{2} - 5 x^{4 - 10}$

Step 2.2.6.3

Subtract $10$ from $4$ .

$3 x^{2} - 5 x^{- 6}$

Step 2.3

Simplify.

Tap for more steps...

Step 2.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 x^{2} - 5 \frac{1}{x^{6}}$

Step 2.3.2

Combine terms.

Tap for more steps...

Step 2.3.2.1

Combine $- 5$ and $\frac{1}{x^{6}}$ .

$3 x^{2} + \frac{- 5}{x^{6}}$

Step 2.3.2.2

Move the negative in front of the fraction.

$f'' (x) = 3 x^{2} - \frac{5}{x^{6}}$